Analytic combinatorics of chord and hyperchord diagrams with $k$ crossings

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diagrams with k crossings

Vincent Pilaud, Juanjo Rué

To cite this version:

Vincent Pilaud, Juanjo Rué. Analytic combinatorics of chord and hyperchord diagrams withkcross- ings. 25th International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA’14), Jun 2014, Paris, France. pp.339-350. �hal-02353375�

Analytic combinatorics of chord and hyperchord diagrams with k crossings

Vincent Pilaud^1∗, and Juanjo Ru´e^2‡

1CNRS & LIX, ´Ecole Polytechnique, France

2Institut f¨ur Mathematik und Informatik, Freie Universit¨at Berlin, Germany

Abstract.Using methods from Analytic Combinatorics, we study the families of perfect matchings, partitions, chord diagrams, and hyperchord diagrams on a disk with a prescribed number of crossings. For each family, we express the generating function of the configurations with exactlykcrossings as a rational function of the generating function of crossing-free configurations. Using these expressions, we study the singular behavior of these generating functions and derive asymptotic results on the counting sequences of the configurations with preciselykcrossings. Limiting distributions and random generators are also studied.

Keywords:Quasi-Planar Configurations — Chord Diagrams — Analytic Combinatorics — Generating Functions

1 Introduction

Quasi-planar chord diagrams. LetV be a set ofnpoints on the unit circle. Achord diagramonV is a set of chords between points ofV. We say that two chordscrosswhen their relative interior intersect.

Thecrossing graphof a chord diagram is the graph with a vertex for each chord and an edge between any two crossing chords. The enumeration properties of crossing-free (or planar) chord diagrams have been widely studied in the literature, see in particular the results of P. Flajolet and M. Noy in [FN99]. From the work of J. Touchard [Tou52] and J. Riordan [Rio75], we know a remarkable explicit formula for the distribution of crossings among all perfect matchings, which was exploited in [FN00] to derive, among other parameters, the limit distribution of the number of crossings for matchings with many chords.

A more recent trend studies chord diagrams with some but restricted crossings. The several ways to restrict their crossings lead to various notions ofquasi-planar chord diagrams. Among others, it is interesting to study chord diagrams

(i) with at mostkcrossings, or

(ii) with no(k+ 2)-crossing (meaningk+ 2pairwise crossing edges), or (iii) where each chord crosses at mostkother chords, or

(iv) which become crossing-free when removing at mostkwell-chosen chords.

∗Supported by the Spanish MICINN grant MTM2011-22792, and by the French ANR grant EGOS 12 JS02 002 01.

‡Supported by JAE-DOC (CSIC), MTM2011-22851 (Spain), and FP7-PEOPLE-2013-CIG projectCountGraph(ref. 630749).

We are grateful to two anonymous referees for helpful comments, corrections, and suggestions.

1365–8050 c⃝2014 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

These conditions are natural restrictions on the crossing graphs of the chord diagrams. Namely, the cross- ing graphs have respectively (i) at mostkedges, (ii) no(k+ 2)-clique, (iii) vertex degree at mostk, and (iv) a vertex cover of sizek. Fork= 0, all these conditions coincide and lead to crossing-free chord dia- grams.

Families of(k+ 2)-crossing-free chord diagrams have been studied in recent literature. On the one hand,(k+2)-crossing-free matchings (as well as their(k+2)-nesting-free counterparts) were enumerated in [CDD⁺07]. On the other hand, maximal(k+ 2)-crossing-free chord diagrams, also called(k+ 1)- triangulations, were enumerated in [Jon05], see also [PS09]. As far as we know, Conditions (i), (iii) and (iv), as well as other natural notions of quasi-planar chord diagrams, still remain to be studied in details.

We focus in this paper on the Analytic Combinatorics of chord diagrams under Condition (i).

Rationality of generating functions. In this paper, we study enumeration and asymptotic properties for different families of configurations: perfect matchings, partitions, chord diagrams, and hyperchord diagrams (our results also extend to partitions and hyperchord diagrams with prescribed block sizes, see [PR13]). Examples of these configurations are represented in Figure 1.

matching partition chord diagram hyperchord diagram

Fig. 1:The four families of (hyper)chord configurations studied in this paper. Their cores are highlighted in bold red.

LetCdenote one of these families of configurations. To avoid handling symmetries, we insert a root in each configuration between two consecutive vertices, and we consider two rooted configurationsCandC^′ ofCas equivalent if there is a continuous bijective automorphism of the circle which sends the root, the vertices, and the (hyper)chords ofC to that ofC^′. We focus on three parameters of the configurations of C: their number nof vertices, their number mof (hyper)chords, and their number kof crossings.

Note that for hyperchord diagrams and partitions, we count all crossings involving two chords contained in two distinct hyperchords. We denote byC(n, m, k)the set of configurations inCwithnvertices,m (hyper)chords andkcrossings, and we letC_k(x, y)^:=!

n,m∈N|C(n, m, k)|xⁿy^mdenote the generating function of the configurations inCwith preciselykcrossings. Our first result concerns the rationality of this function.

Theorem 1 The generating functionCk(x, y)of configurations inCwith exactlykcrossings is a rational function of the generating functionC₀(x, y)of planar configurations inCand of the variablesxandy.

The idea behind this result is to confine crossings of the configurations ofCto finite subconfigurations.

Namely, we define thecore configurationC^⋆of a configurationC∈Cto be the subconfiguration formed by all (hyper)chords ofCinvolved in at least one crossing. The key observation is that

(i) there are only finitely many core configurations withkcrossings, and

(ii) any configurationC ofC withkcrossings can be constructed from its core configurationC^⋆ by inserting crossing-free subconfigurations in the connected components of the complement ofC^⋆in the disk.

This translates in the language of generating functions to a rational expression ofC_k(x, y)in terms ofC₀(x, y)and its successive derivatives with respect tox, which in turn are rational inC₀(x, y)and the variablesxandy. Similar decomposition ideas were used for example by E. Wright in his study of graphs with fixed excess [Wri77, Wri78], or more recently by G. Chapuy, M. Marcus, G. Schaeffer in their enumeration of unicellular maps on surfaces [CMS09].

Note that Theorem 1 extends a specific result of M. B´ona [B´on99] who proved that the generating function of the partitions withkcrossings is a rational function of the generating function of the Catalan numbers. We note that his method was slightly different. The advantage of our decomposition scheme is to be sufficiently elementary and general to apply to the various families of configurations mentioned above.

Asymptotic analysis and random generation. From the expression of the generating functionC_k(x, y) in terms ofC0(x, y), we can extract the asymptotic behavior of configurations inCwithkcrossings.

Theorem 2 Fork≥1, the number of configurations inCwithkcrossings andnvertices is [xⁿ]C_k(x,1) =

n→∞Λn^αρ⁻ⁿ(1 +o(1)),

for certain constantsΛ,α,ρ∈Rdepending on the familyCand on the parameterkas follows.

family constantΛ exponentα singularityρ⁻¹ Prop.

matchings⁽ⁱ⁾

√2 (2k−3)!!

4^k−1k!Γ"

k− ¹2

# k− 3

2 2 11

partitions (2k−3)!!

2^3k−1k!Γ"

k− ¹₂# k− 3

2 4 13

chord diagrams

"

−2 + 3√ 2#3k$

−140 + 99√

2 (2k−3)!!

2^3k+1(3−4√

2)^k−1k!Γ(k−¹₂) k− 3

2 6 + 4√

2 17

hyperchord

diagrams⁽ⁱⁱ⁾ ≃ 1.034^3k0.003655 (2k−3)!!

0.03078^k−1k!Γ(k− ¹₂) k− 3

2 ≃64.97 21

From the rational expression of the generating functionC_k(x, y), we also derive random generation schemes for the configurations inCwith preciselykcrossings, using the methods developed in [DFLS04].

Overview. Due to space limitation, we have decided to present the detailed analysis only for perfect matchings withkcrossings, since we believe that it already gives the flavor of our results and illustrates our method, while remaining technically elementary. For the remaining families, we only report our results and skip the detailed analysis. We refer the interested reader to the long version of this paper [PR13].

(i)The asymptotic estimate for the number of matchings withnvertices is obviously only valid whennis even.

(ii)The expression ofρ⁻¹andΛfor hyperchord diagrams is obtained from approximations of roots of polynomials, and approximate evaluations of analytic functions. Details can be found in Propositions 19 and 21.

2 Perfect matchings

2.1 Perfect matchings and their cores

In this section, we consider the familyMof perfect matchings with endpoints on the unit circle. Each perfect matchingM ofM isrooted: we mark (with the symbol △) an arc of the circle between two endpoints ofM. See Figure 1 (left). LetM(n, k)denote the set of matchings inMwithnvertices andk crossings. We denote byM_k(x)^:=!

n∈N|M(n, k)|xⁿ the generating function of perfect matchings with exactlykcrossings, wherexencodes the number of vertices.

LetM be a perfect matching with some crossings. Our goal is to isolate the contribution of the chords ofM involved in crossings from that of the chords ofM with no crossings.

Definition 3 Acore matchingis a perfect matching where each chord is involved in a crossing. It is a k-core matchingif it has exactlykcrossings. ThecoreM^⋆of a perfect matchingMis the submatching ofM formed by all its chords involved in at least one crossing. See Figure 1 (left).

LetK be a core matching. We letn(K)denote its number of vertices andk(K)denote its number of crossings. We callregionsofKthe connected components of the complement ofK in the unit disk.

A region hasiboundary arcsif its intersection with the unit circle hasiconnected arcs. We letni(K) denote the number of regions ofKwithiboundary arcs, and we setn(K)^:= (ni(K))i∈[k(K)](all regions ofK have at mostk(K)boundary arcs). Since a crossing only involves2chords, ak-core matching can have at most2kchords. This immediately implies that there are only finitely manyk-core matchings.

Definition 4 We encode the finite list of all possible k-core matchingsK and their parametersn(K) andn(K)^:= (ni(K))i∈[k]in thek-core matching polynomial

KM_k(x)^:=KM_k(x1, . . . , xk)^:= %

K k-core matching

x^n(K)

n(K) ^:= %

K k-core matching

1 n(K)

&

i∈[k]

xini(K).

Example 5 The7-core of the matching of Figure 1 (left) contributes toKM₇(x)as₂₄¹ x117x22x3. Remark 6 There is an efficient enumeration algorithm to generate all connected matchings (i.e. whose crossing graph is connected), from which we can easily compute thek-core matching polynomialKM_k(x).

We refer to [PR13, Sections 2.2 and 2.3] for details on this algorithm.

2.2 Generating function of matchings withkcrossings

We study perfect matchings withkcrossings focussing on theirk-cores. For this, we consider the fol- lowing weaker notion of rooting of perfect matchings. We say that a perfect matching withkcrossings is weakly rootedif we have marked an arc between two consecutive vertices of itsk-core. Note that a rooted perfect matching is automatically weakly rooted (the weak root marks the arc of thek-core containing the root of the matching), while a weakly rooted perfect matching corresponds to several rooted perfect matchings. To overcome this technical problem, we use the following rerooting argument.

Lemma 7 LetKbe ak-core withn(K)vertices. The numberMK(n)of rooted matchings onnvertices with coreKand the numberM¯K(n)of weakly rooted matchings onnvertices with coreKare related by

n(K)MK(n) =nM¯K(n).

Observe now that we can construct any perfect matching withkcrossings by inserting crossing-free submatchings in the regions left by itsk-core. From thek-core matching polynomialKM_k(x), we can therefore derive the following expression of the generating functionM_k(x).

Proposition 8 For any k ≥ 1, the generating functionM_k(x)of the rooted perfect matchings withk crossings is given by

Mk(x) =x d dxKMk

'

xi← xⁱ (i−1)!

dⁱ⁻¹ dxⁱ⁻¹

"

xⁱ⁻¹M0(x)#( . In particular,Mk(x)is a rational function ofM0(x)andx.

Proof:Consider a rooted crossing-free perfect matchingM. We say thatMisi-markedif we have placed i−1additional marks between consecutive vertices ofM. Note that we can place more than one mark between two consecutive vertices but that the root is always distinguishable from the other marks. Since we have"n+i−1

i−1

#possible ways to place these(i−1)additional marks, the generating function of the i-marked crossing-free perfect matchings is given by

1 (i−1)!

dⁱ⁻¹ dxⁱ⁻¹

"

xⁱ⁻¹M₀(x)# .

Consider now a weakly rooted perfect matchingM withk≥1crossings. We decompose this matching into several submatchings as follows. On the one hand, the coreM^⋆contains all crossings ofM. This core is rooted by the root ofM. On the other hand, each regionRofM^⋆contains a (possibly empty) crossing-free submatchingMR. We root this submatchingMRas follows:

(i) if the root ofM is not the regionR, thenMRis just rooted by the root ofM;

(ii) otherwise,MRis rooted on the boundary arc ofM^⋆just before the root ofMin clockwise direction.

Moreover, we place additional marks on the remaining boundary arcs of the complement ofRin the unit disk. We thus obtain a rootedi-marked crossing-free submatchingMR in each regionR ofM^⋆with iboundary arcs. Conversly, we can reconstruct the weakly rooted perfect matchingM from its rooted coreM^⋆and its rootedi-marked crossing-free submatchingsMR.

By this bijection, we thus obtain the generating function of weakly rooted perfect matchings with kcrossings. From this generating function, and by application of Lemma 7, we derive the generating functionM_k(x)of rooted perfect matchings withkcrossings:

M_k(x) = %

K k-core matching

x n(K)

d

dxx^n(K)&

i≥1

' 1

(i−1)!

dⁱ⁻¹ dxⁱ⁻¹

"

xⁱ⁻¹M₀(x)#(ni(K)

(1)

=x d dxKM_k

'

xi← xⁱ (i−1)!

dⁱ⁻¹ dxⁱ⁻¹

"

xⁱ⁻¹M₀(x)#( .

SinceM0(x) = ^1−√_2x¹⁻2^4x2 satisfies the functional equationM0(x) = 1+x²M0(x)², all its successive

derivatives, and thusM_k(x), are rational inM₀(x)andx. ✷

Example 9 SinceKM₁(x) =¹₄x14, we obtain the coefficients ofM₁(x)(see Seq. A002694 in OIES):

M1(x) = x⁴M₀(x)⁴ 1−2x²M0(x)=

"

1−√

1−4x²#4

16x⁴√

1−4x² =x⁴+ 6x⁶+ 28x⁸+ 120x¹⁰+ 495x¹²+ 2002x¹⁴. . .

2.3 Asymptotic analysis

We now describe the asymptotic behavior of the number of perfect matchings with k ≥ 1crossings.

The method consists in studying the asymptotic behavior ofM₀(x)and of all its derivatives around their dominant singularities, and to exploit the rational expression ofM_k(x)in terms ofM₀(x)andxgiven in Proposition 8. Along the way, we naturally study whichk-cores have the main asymptotic contributions.

For that, we need the following lemma, whose detailed proof can be found in [PR13, Lemma 2.13].

Lemma 10 The following assertions are equivalent for an (unrooted)k-core matchingK:

(i) Kbelongs to the family of thek-core matchings whose first five elements are shown in Figure 2.

(ii) n1(K) = 3k,nk(K) = 1andni(K) = 0for all other values ofi(here,k≥2).

(iii) Kmaximizesn1(K)among all possiblek-core matchings (here,k≥3).

(iv) Kmaximizes the potentialφ(K)^:=!

i>1(2i−3)ni(K)among all possiblek-core matchings.

We call ak-core matchingmaximalif it satisfies these conditions.

Fig. 2:Maximal core matchings (unrooted) fork= 1, . . . ,5.

Proposition 11 For anyk≥1, the number of perfect matchings withkcrossings andn= 2mvertices is [x^2m]M_k(x) =

m→∞

(2k−3)!!

2^k−1k!Γ"

k−¹₂#m^k−324^m(1 +o(1)),

where(2k−3)!!^:= (2k−3)·(2k−5)· · ·3·1and(−1)!! = 1.

Proof:We assume here thatk≥2(the casek= 1can be derived from a direct analysis of the expression ofM1(x)given in Example 9). We first study the asymptotic behavior ofM0(x)and of all its derivatives around their dominant singularities. The generating functionM₀(x)defines an analytic function around the origin. Its dominant singularities are located atx =±¹₂. Denoting byX+ :=√

1−2x, the singular expansions ofM₀(x)and its derivative aroundx= ¹₂are

M₀(x) =

x∼¹2

2−2√

2X++O"

X+2#

, d

dxM₀(x) =

x∼¹2

2√

2X+−1 +O(1),

and dⁱ

dxⁱM0(x) =

x∼¹2

2√

2 (2i−3)!!X+1−2i +O"

X+2−2i#

for alli≥2,

where(2i−3)!!^:= (2i−3)·(2i−5)· · ·3·1. These expansions are valid in a dented domain atx= ¹₂, [FS09].

We now exploit the expression of the generating functionM_k(x)given by Equation (1) in the proof of Proposition 8. The dominant singularities ofM_k(x)are located atx=±¹₂. We provide the full analysis

aroundx= ¹₂, the computation forx=−¹₂being similar. For conciseness in the following expressions, we set by convention(−1)!! = 1. We therefore obtain:

M_k(x) =x d dx

%

K k-core matching

1 n(K)

&

i≥1

' xⁱ (i−1)!

dⁱ⁻¹ dxⁱ⁻¹

"

xⁱ⁻¹M₀(x)#(ni(K)

=

x∼¹2

1 2

d dx

%

K k-core matching

1 n(K)

&

i>1

'√

2 (2i−5)!!

4ⁱ⁻¹(i−1)! X+3−2i+O"

X+4−2i#(ni(K)

x=∼¹2

1 2

%

K k-core matching

φ(K) n(K)

&

i>1

'√2 (2i−5)!!

4ⁱ⁻¹(i−1)!

(ni(K)

X+−φ(K)−2+O)

X+−φ(K)−1* ,

whereφ(K)^:=!

i>1(2i−3)ni(K)is the potential function introduced in Lemma 10. Observe that in order to obtain the second equality, we used the fact that k >1, and thus, that there existsk-coresK such that ni(K) ̸= 0 when i > 1. Combining Lemma 10 and the Transfer Theorem of singularity analysis [FO90], we conclude that the main contribution in the asymptotic of the previous sum arises from maximalk-cores, as they maximize the value2 +φ(K). By Lemma 10, there are exactly 4 rooted maximalk-cores withn1(K) = 3k,nk(K) = 1,n(K) = 4k, andφ(K) = 2k−3. Hence,

[xⁿ]Mk(x) =

x∼¹2

[xⁿ]1 2

%

K k-core matching

φ(K) n(K)

&

i>1

'√2 (2i−5)!!

4ⁱ⁻¹(i−1)!

(ni(K)

X+−φ(K)−2+O)

X+−φ(K)−1*

x=∼¹2

2√

2 (2k−3)!!

4^kk! [xⁿ]√

1−2x^1−2k+O"

(1−2x)^1−k#

n→∞= 2√

2 (2k−3)!!

4^kk!Γ"

k− ¹2

#n^k−322ⁿ(1 +o(1)),

where the last equality is obtained by application of the Transfer Theorem of singularity analysis [FO90].

Finally, we obtain the stated result by adding the expressions obtained when studyingMk(x)around x= ¹₂andx=−¹₂. In fact, one can check that the asymptotic estimate of[xⁿ]M_k(x)aroundx=−¹₂ is the same but with an additional multiplicative constant(−1)ⁿ. Consequently, the contribution is equal to 0whennis odd and to the estimate in the statement whennis even. ✷

2.4 Random generation

The composition scheme presented in Proposition 8 can also be exploited in order to provide Boltz- mann samplers for random generation of perfect matchings with kcrossings. Throughout this section we consider a positive real numberθ< ¹₂, which acts as a “control-parameter” for the random sampler (see [DFLS04] for further details). The Boltzmann sampler works in three steps:

(i) We first decide which is the core of our random object.

(ii) Once this core is chosen, we complete the matching by means of non-crossing matchings.

(iii) Finally, we place the root of the resulting perfect matching withkcrossings.

We start with the choice of thek-core. For eachk-coreK, letM_K(x)denote the generating function of matchings withkcrossings and whosek-core isK, wherexmarks as usual the number of vertices. Note that this generating function is computed as in Proposition 8, using only the contribution of thek-coreK. Therefore, we have

M_k(x) = %

K k-core matching

M_K(x).

This sum defines a probability distribution as follows: once fixed the parameterθ, letpK= ^M_M^K_k(θ)^(θ). This set of values defines a discrete probability distribution{pK}^{K k-core}_matching, which can be easily simulated.

Once we have fixed the core of the random matching, we continue in the second step filling in its regions with crossing-free perfect matchings. For this purpose it is necessary to have a procedure to generate crossing-free perfect matchings, namelyΓM₀(θ). As M₀(θ)satisfies the recurrence relation M₀(θ) = 1 +θ²M₀(θ)², a Boltzmann sampler ΓM₀(θ)can be defined in the following way. Let p= _ΓM¹_0(θ). Then, using the language of [DFLS04],

ΓM₀(θ)^:= Bern(p)−→∅|(ΓM₀(θ),•−•,ΓM₀(θ)),

where•−•means that the Boltzmann sampler is generating a single chord (or equivalently, two ver- tices in the border of the circle). This Boltzmann sampler is defined whenθ < ¹₂, in which case the defined branching process is subcritical. In such situation the algorithm stops in finite expected time, see [DFLS04].

Once this random sampler is performed, we can deal with a term of the form_dx^di−1i−1xⁱ⁻¹M₀(θ). Indeed, once a random crossing-free perfect matchingΓM₀(θ)of sizen(ΓM₀(θ))is generated, there exist

'n(ΓM₀(θ)) +i−1 i−1

(

i-marked crossing-free perfect matchings arising fromΓM0(θ). Hence, with uniform probability we can choose one of thesei-marked crossing-free perfect matchings. As this argument follows for each choice ofi, andKM_K(x)is a polynomial, we can combine the generator ofi-marked crossing-free diagrams with the Boltzmann sampler for the cartesian product of combinatorial classes (recall that we need to provide the substitutionxi← _(i−^xi_1)!dx^di−1i−1"

xⁱ⁻¹M₀(x)# ).

Finally, we need to apply the root operator, which can be done by means of similar arguments as in the case ofi-marked crossing-free diagrams.

Concerning the statistics of the random variableNcorresponding to the size of the element generated by means of the previous random sampler, as it is shown in [DFLS04], the expected valueE[N]and the varianceVar[N]of the random variableNsatisfy

E[N] =θM^′_k(θ)

M_k(θ) and Var[N] = θ²(M^′′_k(θ)Mk(θ)−θM^′_k(θ)²) +θM^′_k(θ) M_k(θ)² . Hence, whenθtends to ¹₂, the expected value of the generated element tends to infinity, and the variance for the expected size also diverges.

3 Extension to other families of chord diagrams

3.1 Partitions

We first extend our results to the family P of partitions of point sets on the unit circle. See Figure 1.

As before, the partitions are rooted by a mark on an arc between two vertices. Acrossingbetween two blocksU, V of a partitionP is a pair of crossing chordsu1u2andv1v2whereu1, u2∈Uandv1, v2∈V. We count crossings with multiplicity: two blocksU, V cross as many times as the number of such pairs of crossing chords amongUandV. Note that perfect matchings are particular partitions where all blocks have size 2. Applying the same method as in Section 2.2, we obtain an expression of the generating functionPk(x, y)of partitions withkcrossings in terms of thek-core partition polynomialKPk(x, y).

Proposition 12 For anyk≥1, the generating functionP_k(x, y)of partitions withkcrossings is given by Pk(x, y) =x d

dxKPk

'

xi← xⁱ (i−1)!

dⁱ⁻¹ dxⁱ⁻¹

"

xⁱ⁻¹P0(x, y)# , y

( . In particular,P_k(x, y)is a rational function ofP₀(x, y)andx.

From this expression, we can extract as in Section 2.3 asymptotic estimates for the number of partitions withkcrossings, and we can as well construct efficient random generators as in Section 2.4.

Proposition 13 For anyk≥1, the number of partitions withkcrossings andnvertices is [xⁿ]P_k(x,1) =

n→∞

(2k−3)!!

2^3k−1k!Γ"

k−¹₂#n^k−324ⁿ(1 +o(1)).

Remark 14 Our results on matchings and partitions can even be extended to analyze the generating functionP^S_k(x, y)of partitions withkcrossings and whose block sizes all belong toS. In contrast to Proposition 12 which can be directly adapted to this context, the asymptotic analysis ofP^S_k(x, y)involves more technical tools, including the theory of A. Meir and J. Moon on the singular behavior of generating functions defined by a smooth implicit function schema [MM89]. See [PR13, Section 2.10].

3.2 Chord diagrams

We now consider the familyDof all chord diagrams on the unit circle. Remember that a chord diagram is given by a set of vertices on the unit circle, and a set of chords between them. In particular, we allow isolated vertices, as well as several chords incident to the same vertex, but not multiple chords with the same two endpoints. We are interested in the generating functionD_k(x, y)of chord diagrams withk crossings. The generating functionD₀(x, y)of crossing-free chord diagrams was studied in [FN99].

Proposition 15 ([FN99, Equation (22)]) The generating functionD₀(x, y)of crossing-free chord dia- grams satisfies the functional equation

yD₀(x, y)²+"

x²(1 +y)−x(1 + 2y)−2y#

D₀(x, y) +x(1 + 2y) +y= 0.

Therefore, all derivatives _dx^diiD₀(x, y)are rational functions inD₀(x, y)andx. Moreover, we have D₀(x,1) =

x∼ρα−β$

1−ρ⁻¹x+O(1−ρ⁻¹x), where ρ⁻¹= 6 + 4√

2, α=−1 + 3

√2

2 , and β= 1 2

+

−140 + 99√ 2.

As for matchings, we can construct any chord diagram withk crossings by inserting crossing-free subdiagrams in the regions left by itsk-core. We can therefore derive the following expression for the generating functionD_k(x, y)of diagrams withkcrossings, in terms of the generating functionD₀(x, y) of crossing-free diagrams, of thek-core diagram polynomialKD_k(x, y), and of the polynomials

Dⁿ₀(y)^:= [xⁿ]D₀(x, y) and D^≤₀^p(x, y)^:= %

n≤p

Dⁿ₀(y)xⁿ= %

n≤p m≥0

|D(n, m,0)|xⁿy^m. Proposition 16 For anyk≥1, the generating functionD_k(x, y)of chord diagrams withkcrossings is given by

D_k(x, y) =x d dxKD_k

,

x0,j ←D^j₀(y)

x^j , xi,j← xⁱ (i−1)!

dⁱ⁻¹ dxⁱ⁻¹

D₀(x, y)−D^≤₀^i+j(x, y)

x^i+j+1 , y

- . In particular,Dk(x, y)is a rational function ofD0(x, y)andx.

Similarly to our asymptotic analysis in Section 2.3, we can obtain asymptotic results for the number of chord diagrams withkcrossings, in terms of the constantsρ,α, andβdefined in Proposition 15. Random generators can as well be constructed, see [PR13, Section 3.6].

Proposition 17 For anyk≥1, the number of chord diagrams withkcrossings andnvertices is [xⁿ]Dk(x,1) =

n→∞

α^3kβ(2k−3)!!

(2ρ)^k−1k!Γ(k− ¹₂)n^k−32ρ⁻ⁿ(1 +o(1)).

3.3 Hyperchord diagrams

As from matchings to partitions, we can finally extend our results from chord diagrams to hyperchord diagrams. Ahyperchordis the convex hull of finitely many points of the unit circle. Given a point setV on the circle, ahyperchord diagramonV is a set of hyperchords with vertices inV. Note that we allow isolated vertices in hyperchord diagrams. As for partitions, acrossingbetween two hyperchordsU, V is a pair of crossing chordsu1u2andv1v2, withu1, u2 ∈U andv1, v2∈V. We consider the familyHof hyperchord diagrams, and we want to analyze the generating functionH_k(x, y)of hyperchord diagrams withkcrossings, counted with multiplicities. As for chord diagrams, our first step is to study the gen- erating functionH₀(x, y)of crossing-free hyperchord diagrams, extending the results of P. Flajolet and M. Noy for chord diagrams [FN99] that we presented in Proposition 15.

Proposition 18 The generating function H₀(x, y) of crossing-free hyperchord diagrams satisfies the functional equation

p3(x, y)H₀(x, y)³+p2(x, y)H₀(x, y)²+p1(x, y)H₀(x, y) +p0(x, y) = 0, where

p0(x, y)^:= −2x²−x+ 2x y³+y²+x²y⁴−7x²y−7x²y²−x²y³−3x y, p1(x, y)^:= −2x³−2x³y⁴−8x³y+ 2x−3y²−12x³y²−8x³y³

+ 6x y−x²y⁴+x²+ 4x²y+ 4x²y²−4x y³, p2(x, y)^:=x²y³+x²+ 3x²y²−x−3x y+ 2x y³+ 3x²y+ 3y², p3(x, y)^:= −y².